In this thesis, we focus on the study of hyperkahler metric in four dimensional cases, and practice GMN's construction of hyperkahler metric on focus-focus fibrations. We explicitly compute the action-angle coordinates on the local model of focus-focus fibration, and show its semi-global invariant should be harmonic to admit a compatible holomorphic 2-form. Then we study the canonical semi-flat metric on it. After the instanton correction inspired by physics, we get a family of the generalized Ooguri- Vafa metric on focus-focus fibrations, which becomes more local examples of explicit hyperkahler metric in four dimensional cases. In addition, we also make some exploration of the Ooguri-Vafa metric in the thesis. We study the potential function of the Ooguri-Vafa metric, and prove that its nodal set is a cylinder of bounded radius 1 < R < 1. As a result, we get that only on a finite neighborhood of the singular fibre the Ooguri-Vafa metric is a hyperkahler metric. Finally, we give some estimates for the diameter of the fibration under the Oogui-Vafa metric, which confirms that the Oogui-Vafa metric is not complete. The new family of metric constructed in the thesis, we think, will provide more examples to further study of Lagrangian fibrations and mirror symmetry in future.